Convexity Correction¶

When pricing derivatives we work under a risk-neutral measure and require that the discounted asset price is a martingale. If we model the log-return directly as a stochastic process \(x_t\), that process will generally not satisfy this condition on its own. The convexity correction is the deterministic adjustment that restores it.

Setup¶

Consider an asset price of the form

\[\begin{equation} S_t = S_0\, e^{s_t} \end{equation}\]

where \(s_t\) is the log-return. We model \(s_t\) as

\[\begin{equation} s_t = x_t - c_t \end{equation}\]

where \(x_t\) is the driving stochastic process (e.g. a Brownian motion, a Lévy process, or a stochastic-volatility process) and \(c_t\) is a deterministic function of time.

The no-arbitrage condition (with zero interest rates) requires the forward price to equal the spot price, i.e. \({\mathbb E}[S_t] = S_0\), which is equivalent to

\[\begin{equation} {\mathbb E}\!\left[e^{s_t}\right] = 1 \end{equation}\]

The Correction Term¶

Substituting \(s_t = x_t - c_t\) into the martingale condition gives

\[\begin{align*} {\mathbb E}\!\left[e^{x_t - c_t}\right] &= 1 \\ e^{-c_t}\,{\mathbb E}\!\left[e^{x_t}\right] &= 1 \end{align*}\]

so the correction must satisfy

\[\begin{equation} c_t = \log {\mathbb E}\!\left[e^{x_t}\right] \end{equation}\]

In terms of the characteristic exponent \(\phi_{x_t, u}\) (defined by \(\Phi_{x_t}(u) = e^{-\phi_{x_t,u}}\)), evaluating at \(u = -i\) gives

\[\begin{equation} {\mathbb E}\!\left[e^{x_t}\right] = \Phi_{x_t}(-i) = e^{-\phi_{x_t,-i}} \end{equation}\]

therefore

\[\begin{equation} c_t = -\phi_{x_t, -i} \end{equation}\]

This is the log-Laplace transform (cumulant generating function) of \(x_t\) evaluated at 1. It is always real-valued and non-negative by Jensen's inequality, since \(e^{x_t}\) is convex (hence the name convexity correction).

Effect on the Characteristic Function¶

The characteristic function of the log-return \(s_t = x_t - c_t\) is

\[\begin{equation} \Phi_{s_t}(u) = {\mathbb E}\!\left[e^{iu s_t}\right] = {\mathbb E}\!\left[e^{iu(x_t - c_t)}\right] = e^{-iuc_t}\,\Phi_{x_t}(u) \end{equation}\]

The correction introduces a phase shift proportional to \(c_t\). Two key values confirm the martingale property:

\(\Phi_{s_t}(0) = 1\) (normalization, always true)
\(\Phi_{s_t}(-i) = e^{-c_t}\,{\mathbb E}[e^{x_t}] = e^{-c_t} e^{c_t} = 1\) (martingale condition)

Wiener Process¶

For a Brownian motion \(x_t = \sigma W_t\) the moment generating function is

\[\begin{equation} {\mathbb E}\!\left[e^{x_t}\right] = e^{\sigma^2 t / 2} \end{equation}\]

so the correction is

\[\begin{equation} c_t = \frac{\sigma^2 t}{2} \end{equation}\]

This is the classical term that appears in the Black-Scholes formula. It grows linearly with time, reflecting the fact that geometric Brownian motion drifts upward on average (the asset price is log-normally distributed with positive variance).

from quantflow.sp.weiner import WeinerProcess
pr = WeinerProcess(sigma=0.5)
-pr.characteristic_exponent(1, complex(0,-1))  # c_t at t=1

which is the same as

pr.convexity_correction(1)

General Lévy Process¶

For any Lévy process \(x_t\) with characteristic exponent \(\psi\) (so that \(\phi_{x_t, u} = t\,\psi(u)\)), the correction is

\[\begin{equation} c_t = -t\,\psi(-i) = t\,\log {\mathbb E}\!\left[e^{x_1}\right] \end{equation}\]

The correction is linear in time for all Lévy processes. For processes with jumps (e.g. Merton jump-diffusion, variance gamma) the jump component contributes an additional positive term on top of the diffusion \(\sigma^2 t / 2\).